#506179

order	512
nilpotency class	4
isoclinism	isoclinic class of #506179
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#28080
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1765 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,8,15,9,16,7,14)(3,12,20,45,21,47,36,46)(4,22,27,11,28,43,26,44)(5,29,31,39,32,38,19,10)(6,33,17,13,35,49,30,50)(18,52,23,59,55,57,54,41)(24,37,56,40,61,63,53,62)(25,42,51,64,34,48,60,58);(1,3,17,5,8,20,35,31,9,21,30,32,7,36,6,19)(2,10,33,46,14,38,50,47,16,39,49,45,15,29,13,12)(4,23,60,53,26,18,34,61,28,54,51,56,27,55,25,24)(11,40,64,41,43,63,48,52,44,62,58,59,22,37,42,57);(1,4)(2,11)(3,18)(5,24)(6,34)(7,27)(8,26)(9,28)(10,37)(12,41)(13,48)(14,43)(15,22)(16,44)(17,51)(19,56)(20,54)(21,55)(23,36)(25,35)(29,62)(30,60)(31,53)(32,61)(33,58)(38,40)(39,63)(42,50)(45,57)(46,52)(47,59)(49,64);(1,5,30,20,8,31,6,21,9,32,17,36,7,19,35,3)(2,12,33,39,14,46,50,29,16,47,49,10,15,45,13,38)(4,24,60,54,26,53,34,55,28,61,51,23,27,56,25,18)(11,41,58,63,43,52,42,62,44,59,64,37,22,57,48,40)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,d^2=ef,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=dh,f^b=fi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413375);